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236                       ANALYTICAL MECHANICS
4.  The return trough of a bowling alley is 50 feet long and has a slope of 1 foot in 20 feet.   Supposing the contact to be perfectly rough find the time a ball will take to return.   The sides
of the trough are perpendicular to each other.
5.  In the adjoining figure the largest circle represents a solid disk wheel, which rolls along a rough horizontal table under the action of a falling body.   The left-hand end of the string is spliced and connected to two smooth rings on the axle of the wheel.
The pulley over which the string passes is smooth.   Discuss the motion.
6.  In the preceding problem suppose the pulley to be rough and to rotate about its axis.
7.  Same as problem 5 except that the wheel rolls up an inclined plane.
8.  In the preceding problem suppose the pulley to rotate.
9.  Same problem as the third illustrative example, p. 231, except that the cylinder is hollow and has a negligible thickness.
10.   Same as the preceding problem, but the cylinder rolls up an inclined plane.
11.  How can you tell a solid sphere from a hollow one which has exactly the same diameter and mass?
12.  Two men of different weights coast down a hill on exactly similar bicycles.   Which will reach the bottom of the hill first, the lighter or the heavier man?
13.  A thin spherical shell of perfectly smooth inner surface is filled with water and allowed to roll down an inclined plane.   Discuss the motion.
14.  A hollow cylinder of negligible thickness and perfectly smooth inner surface is filled with water and allowed to roll down an inclined plane.   Discuss the motion.
GENERAL PROBLEMS.
1.  A sphere of radius a starts from the top of a fixed sphere of radius b and rolls down.   If there is no sliding find the position at which they will separate.
2.  Two masses Wi and w2 are suspended by means of strings which are wound around a wheel and its axle, respectively.   The wheel and axle are rigidly connected and are free to rotate about a horizontal axis.   Discuss the motion,